General fractional calculus approach to a three-dimensional nonlinear reaction–diffusion system for atherosclerosis: LDL, Lipoprotein(a) and CRP dynamics with Sonine kernels

We develop a three-dimensional nonlinear reaction–diffusion model for the biochemical interactions of low-density lipoprotein (LDL), lipoprotein(a), and C-reactive protein (CRP) within arterial walls three key biomarkers of atherosclerosis progression. Memory and hereditary effects are incorporated through general fractional derivatives (GFDs) of convolution type with Sonine kernels, in the framework recently formalised by Luchko. Within this setting, the classical Riemann–Liouville and Caputo fractional derivatives appear as particular cases corresponding to the power-function kernel k(t)=h1−α(t), while richer biological memory structures such as two-scale chronic–acute kernels of Mittag-Leffler type are admitted as long as the Sonine condition is satisfied. All admissible kernels in the Sonine class S−1 possess an integrable singularity at the origin, consistent with the structural requirement that follows from the two fundamental theorems of fractional calculus within the Sonine framework; fractional operators built on non-singular kernels form a parallel and actively studied class outside S−1 and are not analysed here. We prove existence and uniqueness of solutions via a fixed-point argument adapted to the Sonine-kernel setting and we derive the associated integral (variation-of-parameters) representation through the two fundamental theorems of fractional calculus. A finite-difference scheme combined with Fourier sine mode decomposition is implemented for the numerical solution; results are reported for the Caputo kernel and for the Mittag-Leffler GFD kernel at fractional orders α∈{0.1,0.5,0.8,0.9,1}. The comparative analysis clarifies which memory structure is biologically appropriate for chronic versus acute inflammatory regimes and shows that the GFD framework provides a flexible and mathematically well-justified extension of single-kernel fractional models for cardiovascular disease.